Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats

11    REGULARIZATION       299

7.    Let G be a locally compact group. For each t e R* , let ju,r = 0 be a positive bounded
measure on G. Suppose that the mapping /K-+JU,, is continuous with respect to the
topology $~2 (in the notation of Section 13.20, Problem 1), and that [JLS +1 = p,s * ju,r
for all s, t in Rt.

(a)    Show that there exists a real number e such that ||)Ltt|| = ect, (Observe that the
mapping t h- ||/u,,|| is lower semicontinuous, and apply Problem 6.)

(b)    Show that, as /-*0, p,t converges (with respect to ^"2) to a Haar measure of
total mass 1 on a compact subgroup of G. (Use (12.15.9) to show that there exists a
sequence (sn) tending to 0 such that the sequence (fjLSn) tends to a limit p in the top-
ology $~2 and that /xr * p, = JJLT = p, * //,, for all t > 0. Deduce that jLts -> JK, as s -0,
and that p * JLC = /x. Complete the proof by using Problem 5 of Section 14.7.)

8.    Let A denote Lebesgue measure on R". Let A be a bounded convex open set in R"
(Section 8.5, Problem 8), and let D(A) = A  A be the set of all x  y where :c, y e A.
The set D(A) is convex, open, symmetric and bounded. For each x = 0 in D(A), let
p(x) be the unique real number, belonging to ]0, 1[, such that p(x)"1^ lies in the
frontier of D(A). If we put p(0) = 0, then p is a continuous function on D(A) (Section
12.14, Problem 12).

(a)    For each x e D(A), show that

A(A n (A + x)) ^ (I - pW)"A(A).
(Observe that if x ^ 0 and p(x)~lx = b  a where a, b E A, then

(1 - p(x))A + p(x)b = (1 - p(x))A + p(x)a + x).

(b)    Show that

(A(A))2-f     (^A*9A

(c)    Deduce from (a) and (b) that

A(A)^f     (1-yc


Prove that this integral is equal to

A(D(A)) f1/z/n~100w^ = !

(Split up the interval  [0, 1] into m parts by means of an increasing sequence
(4)o*m, and consider D(A) as the union of the sets defined by tk < p(x) < tk+l;
then let m -* oo.)
Hence show that

9. Let G be a locally compact group, p a left Haar measure on G, and let V be a sym-
metric compact neighborhood of e in G which does not contain any subgroup other
than {e}.s a universally measurable